Ivancevic_Applied-Diff-Geom

1122 Applied Differential Geometry: A Modern Introductionus consider the Chern–Simons theory as an example. The classical action,∫S CS = d 3 x(A ∧ dA + 2 )M 3 A ∧ A ∧ A , (6.190)3is a topological invariant, which gives the second Chern class of 3–manifoldM 3 . As is easy to find, S CS is not invariant under the topological gaugetransformation, although it is YM gauge invariant. Therefore the quantizationis proceeded by the standard BRST method. This is a general featureof Schwarz–type TQFT.6.5.3.1 Dimensional ReductionFirst, let us recall the SW monopole equations in 4D. We assume that Xhas Spin structure. Then there exist rank two positive and negative spinorbundles S ± . For Abelian gauge theory, we introduce a complex line bundleL and a connection A µ on L. The Weyl spinor M(M) is a section of S + ⊗L(S + ⊗ L −1 ), hence M satisfies the positive chirality condition γ 5 M = M.If X does not have Spin structure, we introduce Spin c structure and Spin cbundles S ± ⊗ L, where L 2 is a line bundle. In this case, M should beinterpreted as a section of S + ⊗ L. Below, we assume Spin structure.Recall that the 4D Abelian SW monopole equations are the followingset of differential equationsF + µν + i 2 Mσ µνM = 0, iγ µ D µ M = 0, (6.191)where F + µν is the self–dual part of the U(1) curvature tensorF µν = ∂ µ A ν − ∂ ν A µ , F + µν = P + µνρσF ρσ , (6.192)while P µνρσ + is the self–dual projector defined byP µνρσ + = 1 ( √ ) gδ µρ δ νσ +22 ɛ µνρσ .Note that the second term in the first equation of (6.191) is also self–dual. On the other hand, the second equation in (6.191) is a twisted Diracequation, whose covariant derivative D µ is given byD µ = ∂ µ + ω µ − iA µ , where ω µ = 1 4 ωαβ µ [γ α , γ β ]is the spin connection 1–form on X.